Central Automorphism Groups Fixing the Center Element-Wise

International Electronic Journal of Algebra Volume 9 (2011) 167-170

CENTRAL AUTOMORPHISM GROUPS FIXING THE CENTER ELEMENT-WISE

S. Heydar Jafari Received: 27 May 2010; Revised: 23 November 2010 Communicated by Sait Halıcıo˘glu

Abstract. Let G be a finite p-group. We find a necessary and sufficient condition on G such that each central automorphism of G fixes the center element-wise.

Mathematics Subject Classiﬁcation (2000): 20D45 , 20D15 Keywords: automorphisms, p-groups, nilpotent groups

1. Introduction

Let G be a ﬁnite group. An automorphism σ of G is said to be central if and only if it induces the identity on G/Z(G), or equivalently, g−1σ(g) ∈ Z(G) for all g ∈ G, where Z(G) is the center of G. The central automorphisms of G, denoted by Autc(G), forms a normal subgroup of Aut(G), the group of automorphisms of

G. Notice that Autc(G) = CAut(G)(Inn(G)). For a group H and an abelian group K, Hom(H,K) denotes the group of all homomorphisms from H to K. Let M and N be two normal subgroups of G. By AutN (G) we mean the subgroup of Aut(G) consisting of all automorphisms which induces the identity on G/N. Also by AutM (G) we mean the subgroup of Aut(G) consisting all automorphisms which N N induces identity on M. Let AutM (G) = Aut (G) ∩ AutM (G). we use the notation pn pn G = hg | g ∈ Gi and Ct for cyclic group of order t. The group of central automorphisms of a finite groups is of great importance in the investigation of Aut(G), and has been studied by several authors (see e.g., [1-7]). Z(G) In the article of Attar [2], it is proved for a finite p-group G, AutZ(G)(G) = Inn(G) if and only if G is nilpotent of class 2 and Z(G) is cyclic. M.K.Yadav in [7] obtained Z(G) some necessary and sufficient conditions for the equality Autc(G) = AutZ(G)(G) on p-groups of class 2. We find a necessary and sufficient condition on G in the general case in order Z(G) for Autc(G) = AutZ(G)(G). Z(G) Theorem. Let G be a finite p-group. Then Autc(G) = AutZ(G)(G) if and only 0 n 0 if Z(G)G ⊆ Gp G where exp(Z(G)) = pn . 168 S. HEYDAR JAFARI

2. Proofs

Let G be a finite group and M be a central subgroup of G and σ ∈ AutM (G). −1 Then we can define a homomorphism fσ from G into M such that fσ(x) = x σ(x). On the other hand, given a homomorphism f from G to M, we can always define an endomorphism σf of G such that σf (x) = xf(x). But σf is an automorphism of G if and only if for every non-trivial element x ∈ M, f(x) 6= x−1. A finite group G is said to be purely non-abelian if it has no nontrivial abelian direct factor. In [1], Adney and Yen proved that the correspondence σ 7→ fσ 0 defined above is a one-to-one map of Autc(G) onto Hom(G/G ,Z(G)) for purely non-abelian finite groups. 0 0 We recall that if H ≤ Z(G) and K/G is a direct factor of G/G , then any element 0 0 f of Hom(K/G ,H) induces an element f¯ of Hom(G/G ,H) which is trivial on the 0 0 complement of K/G in G/G . To simplify the notation in the proof of the main theorem, we will identify f with the corresponding homomorphism from G into H. The following Lemma can be obtained from Proposition 2.3 of [6] immediately.

Lemma 2.1. [7, Lemma 2.4] Let G be a ﬁnite non-abelian p-group such that Z(G) Autc(G) = AutZ(G)(G). Then G is purely non-abelian.

Proof. Suppose G has a nontrivial abelian direct factor. Let G = A×H, where A is abelian and H is a non-abelian p-groups. Then Z(H) 6= 1 and so Hom(A, Z(H)) is nontrivial and by [6, Proposition 2.3] it can be assumed to be a subgroup of Autc(G). Z(G) It is clear that Hom(A, Z(H)) 6⊆ AutZ(G)(G), which is a contradiction. ¤

Lemma 2.2. Let G be a ﬁnite abelian p-group and x an element of G. Then, there ps1 pst exist nontrivial elements x1, ..., xt such that G = hx1i×· · ·×hxti and x = x1 ...xt , where si ∈ N ∪ {0}.

r1 rt Proof. Let G = hy1i × · · · × hyti and x = y1 ...yt , where ri ≥ 0. There exist si li li and si such that ri = p li and (p, li) = 1. We have hyii = hyi i and hence l1 lt G = hy1 i × · · · × hyt i. This completes the proof. ¤

Z(G) Lemma 2.3. Let f ∈ Hom(G, Z(G)) and σf ∈ Autc(G). Then σf ∈ AutZ(G)(G) if and only if Z(G) ⊆ Ker(f).

Proof. This follows directly from the deﬁnition of σf . ¤

0 n 0 Lemma 2.4. Let G be a ﬁnite p-group and Z(G)G ⊆ Gp G where exp(Z(G)) = pn. Then G is purely non-abelian. CENTRAL AUTOMORPHISM GROUPS FIXING THE CENTER ELEMENT-WISE 169

Proof. Let G = A × H where A 6= 1 is abelian. Then A ⊆ Z(G) and A is not a n 0 n 0 subset of Gp G = Hp H , which is a contradiction. ¤

Proof of main theorem. By Lemmas 2.1 and 2.4 we can suppose that G is a 0 n 0 purely non-abelian p-group. Let Z(G)G ⊆ Gp G and x ∈ Z(G). Then there exist 0 n elements a in G and b in G such that x = ap b. For any f ∈ Hom(G, Z(G)), since 0 n exp(Z(G)) = pn and G ⊆ Ker(f), we have f(x) = f(a)p f(b) = 1.1 = 1. Hence Z(G) Z(G) ⊆ Ker(f) and σf ∈ AutZ(G)(G). Z(G) 0 pn 0 Conversely, let Autc(G) = AutZ(G)(G) and suppose Z(G)G * G G . So there pn 0 exists x in Z(G) which is not in G G . By Lemma 2.2, there exist x1, ..., xt of G 0 0 0 0 ps1 0 pst 0 such that G/G = hx1G i×· · ·×hxtG i and xG = x1 G ...xt G . Hence for some sj n 0 p p sj n n j, xj is not in G , so p < p . Select z ∈ Z(G), where O(z) = min(O(xjG ), p ), 0 0 and deﬁne f by xjG 7−→ z. Then f can be considered as a homomorphism of G/G 0 ps1 0 pst 0 into Z(G) and so σf ∈ Autc(G). We obtain f(x) = f(xG ) = f(x1 G ...xt G ) = s p j psj psj f(xj ) = z . On the other hand z is a nontrivial element of Z(G). Therefore Z(G) σf is not in AutZ(G)(G), which is a contradiction. In particular it is of interest for groups of class 2 with elementary abelian centers, and generalises a result of M. J. Curran [3, Proposition 2.7].

Corollary 2.5. Let G be a ﬁnite non-abelian p-group and exp(Z(G)) = p. Then Z(G) Autc(G) = AutZ(G)(G) if and only if Z(G) ⊆ φ(G), where φ(G) is frattini subgroup of G.

Lemma 2.6. Let G = Cpa1 × Cpa2 × · · · × Cpat where a1 > a2 > ··· > at and pn H ≤ G for some integer n. If n > ak for some k ∈ {1, ..., t} then, G/H and G ∼ have equal rank, and G/H = (Cpa1 × Cpa2 × · · · × Cpak−1 )/H × (Cpak × · · · × Cpat ). ∼ Moreover (Cpa1 × Cpa2 × · · · × Cpak−1 )/H = Cpn × Cpn × · · · × Cpn if and only if n H = Gp .

The proof immediately follows from the fact that H ∩ (Cpak × · · · × Cpat ) = 1. 0 Let G be a ﬁnite p-group of class 2. Then G/Z(G) and G have equal exponent c p (say). Let G/Z(G) = Cpa1 × Cpa2 × · · · × Cpar , where a1 > a2 > ··· > ar > 0.

Let k be the largest integer between 1 and r such that a1 = a2 = ··· = ak = c. Let 0 a M = M/Z(G) = Cpa1 × Cpa2 × · · · × Cp k . Let G/G = Cpb1 × Cpb2 × · · · × Cpbs , 0 where b1 > b2 > ··· > bs > 0, be a cyclic decomposition of G/G such that M is 0 isomorphic to a subgroup of N = N/G = Cpb1 ×Cpb2 ×· · ·×Cpbk . Using the above terminology, we in particular obtain the main theorem of Yadav [7].

Corollary 2.7. Let G be a ﬁnite non-abelian p-group of class 2. Then the following are equivalent. 170 S. HEYDAR JAFARI

Z(G) (a) Autc(G) = AutZ(G)(G). n 0 (b) Z(G) = Gp G . 0 0 (c) r = s, (G/Z(G))/M =∼ (G/G )/N and the exponent of Z(G) and G are equal.

0 n Proof. Since exp(G/Z(G)) = exp(G ) ≤ exp(Z(G)), we have Gp ⊆ Z(G). There- n 0 fore Gp G ⊆ Z(G), and by main theorem of article (a) and (b) are equivalent. 0 0 Let (b) hold. We ﬁrst show exp(G ) = exp(Z(G)). Let G < Z(G). Then 0 n 0 0 0 Z(G)/G = Gp G /G is nontrivial subgroup of abelian group G/G , so 0 0 exp((G/G )/(Z(G)/G )) = exp(G/Z(G)) = pn. On the other hand exp(G/Z(G)) = 0 0 0 exp(G ) ≤ exp(Z(G)). Thus exp(G ) = exp(Z(G)). The case Z(G) = G , is trivial. Finally (b) and (c) are equivalent by Lemma 2.6. ¤

Acknowledgement: The author would like to thank the referee for his/her very useful comments which improved the paper.

References

[1] J.E. Adney and T. Yen, Automorphims of a p-groups, Illinois J. Math, 9 (1965), 137-143 . [2] M.S. Attar, On central automorphism that fix the central elementwise, Arch. Math., 89 (2007), 296-297. [3] M.J. Curran, Finite groups with Central automorphism group of minimal order, Math. Proc. Royal Irish Acad., 104 A(2) (2004), 223-229. [4] M.H. Jafari, Elementry abelian p-groups as central automorphism groups, Comm. Algebra, 29 (2001), 2081 -2087. [5] M.H. Jafari and A.R. Jamali, On the occurrance of some finite groups in the central automorphism group of finite groups, Math. Proc. Royal Irish Acad., 106 A(2) (2006), 139-148. [6] M.H. Jafari and A.R. Jamali, On the nilpotency and solubility of the central automorphism group of finite group, Algebra Coll., 15:3 (2008), 485-492. [7] Manoj K. Yadav, On central automorphisms fixing the center element-wise, Comm. Algebra, 37 (2009), 4325-4331.

S. Heydar Jafari Department of Mathematics Shahrood University of Technology P. O. Box 3619995161-316, Shahrood, Iran e-mail: [email protected]